(Opinions expressed in this webpage are those of myself, and do not necessarily reflect the view of J.P. Morgan.)
Abstract: The uncertainty in future market dynamics is an important consideration when developing strategies for hedging derivatives, particularly data driven strategies such as deep hedging. Deep market generators can produce higher fidelity training data than classical models, but, like those, typically require frequent recalibration to new market data. The resulting strategies are thus susceptible to underperformance if there is a mismatch (distributional shift) between training data and live data. We present a framework to train a modified deep hedger which displays a form of ambiguity aversion, henceforth termed an Ambiguity-Averse Deep Hedger (AADH). The modeller has full control over exactly which aspects of distributional shifts the AADH is to be robust to, through selection of features relevant to the trading strategy which are used to cluster the training data, allowing for the evaluation of a loss function motivated by the theory of smooth ambiguity aversion. We demonstrate that for forward start options with transaction costs, our AADH with feature clustering always outperforms the standard deep hedger in out-of-distribution tests, emphasising the need to prepare for future uncertainty, and the effectiveness of ambiguity aversion in achieving this. For plain vanilla options with transaction costs, our AADH also outperforms the standard deep hedger in the overwhelming majority of out-of-distribution scenarios. Our findings are supported by numerical experiments involving (i) synthetic data with standard stochastic models, as well as (ii) real data experiments on equity indices over extensive periods of time, as well as (iii) a period of significant distributional shifts, the COVID period.
In plain English: Market reality differs from synthetic market data, and changes over time. Deep hedging, as the name suggests, trains a deep neural network to hedge derivatives. However, deep neural networks are data hungry, and typically require more market data than what we have available. Thus, a synthetic market model is calibrated, and many samples are generated from it, which the deep hedger uses to learn its hedging strategy from. Consequently, the deep hedger learns how to hedge under that synthetic market model. If and when market reality differs from that model, then there is no guarantee that the risk metrics observed are realised.
Our paper describes a proof of concept that this out-of-distribution risk can be kept under control, with the same synthetic data, and with the same samples. We do this by first partitioning the data by clustering features of the samples, such as realised volatility, autocorrelation, or any other quantity which can be computed path-wise. Rather than compute the strategy's risk over the entire dataset, we instead compute the strategy's risk over each feature cluster, and then minimise a "second-order" risk metric (which may be thought of as model risk or Knightian uncertainty). The overall loss function is inspired by the theory of smooth ambiguity aversion.
We deep hedge a call option and a forward start option using a standard deep hedging setup, and using our ambiguity-averse deep hedging setup. What we find is that our ambiguity-averse deep hedger achieves superior performance when the validation data deviates from the synthetic data used to train the models on the majority of real (out-of-distribution) datasets.
How does an ambiguity-averse hedging strategy differ from a plain risk-averse one? When deep hedging a call option with only spot, the effect is very similar to the hedger using a higher implied vol to calculate the hedge ratio. We have intuition about how an ambiguity-averse strategy differs when the option is more exotic, and some intuition about when there are more hedging instruments available, but neither situation was the focus of the paper. Please get in touch if you're thinking of running some experiments like this!
Are we, investors and hedgers, actually ambiguity-averse? There is empirical evidence that people are ambiguity-averse, as described by Ellsberg (1961). Sadly there has not been much investigation into ambiguity appetites in quantitative finance. However, we find that the use of our methodology in deep hedging increases robustness in a very concrete way. (If downstream metrics are all you care about, then our results are evidence that you should be ambiguity-averse.)
TLDR: We have developed a framework, and a proof of concept, for the robustification of deep hedging strategies. This is done by partitioning a market model based on relevant features of market paths, and a loss function which accounts for both first- and second-order risk.
Short gallery: